Last updated: 2026-05-19

Inverse trigonometric functions

Trigonometric functions like sine, cosine, and tangent take an angle and return a ratio. Inverse trig functions reverse that: given a ratio, they return the angle that produces it. If sin(30°) = 0.5, then arcsin(0.5) = 30°. This calculator evaluates the three most common inverse trig functions — arcsin, arccos, and arctan — for any valid input, and returns the result in degrees or radians.

Inverse trig functions appear throughout geometry, physics, and engineering whenever you need to find an unknown angle from known side lengths or signal ratios.

The three principal branches

Each inverse function is defined on a principal branch — a restricted range that makes the output unique:

The domain restriction for arcsin and arccos exists because sine and cosine only produce values between −1 and 1. Tangent has no such limit — it spans all real numbers — so arctan accepts any input.

The principal-value convention

Sine is periodic: sin(30°), sin(150°), sin(390°) all equal 0.5. Infinitely many angles share the same sine, so a strict inverse cannot be defined over the full domain. The principal-value convention resolves this by returning a single, agreed-upon answer drawn from the restricted range. For arcsin that range is the interval [−90°, 90°]; for arccos it is [0°, 180°].

The general formula recovers the remaining solutions. If arcsin(x) = θ, then every solution to sin(α) = x is:

• α = θ + 360°·n
• α = (180° − θ) + 360°·n

where n is any integer.

Worked example

Problem: A right triangle has a hypotenuse of 13 cm and an opposite side of 5 cm. Find the angle opposite the 5 cm side.

sin(θ) = opposite / hypotenuse = 5 / 13 ≈ 0.3846

θ = arcsin(0.3846) ≈ 22.62°

Evaluating arcsin(0.3846) with the output set to degrees returns this angle directly.

Degrees vs radians

Degrees are more intuitive for everyday geometry (a full circle is 360°). Radians are the natural unit in calculus and most programming languages — Math.asin() in JavaScript, asin() in Python, and ASIN() in Excel all return radians.

To convert: degrees = radians × (180/π); radians = degrees × (π/180).